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2019 Finnish National High School Mathematics Comp, 4

Define a sequence $ a_n = n^n + (n - 1)^{n+1}$ when $n$ is a positive integer. Define all those positive integer $m$ , for which this sequence of numbers is eventually periodic modulo $m$, e.g. there are such positive integers $K$ and $s$ such that $a_k \equiv a_{k+s}$ ($mod \,m$), where $k$ is an integer with $k \ge K$.

2013 Middle European Mathematical Olympiad, 5

Tags: bir tinga qimmat masala , geometry , trigonometry

Let $ABC$ be and acute triangle. Construct a triangle $PQR$ such that $ AB = 2PQ $, $ BC = 2QR $, $ CA = 2 RP $, and the lines $ PQ, QR,$ and $RP$ pass through the points $ A, B , $ and $ C $, respectively. (All six points $ A, B, C, P, Q, $ and $ R $ are distinct.)

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